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- W4299626717 abstract "We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{Omega(d)}$ for values of $d = d(n)$ from constant all the way up to $n^{delta}$ for some universal constant$delta$. This shows that the $n^{O(d)}$ running time obtained by using the Lasserre semidefinite programming relaxations to find degree-$d$ SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining $mathsf{NP}$-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Kraj'iv{c}ek '04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, Muller, and Oliva '13] and [Atserias, Lauria, and Nordstrom '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively." @default.
- W4299626717 created "2022-10-02" @default.
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- W4299626717 date "2015-04-07" @default.
- W4299626717 modified "2023-10-16" @default.
- W4299626717 title "Tight Size-Degree Bounds for Sums-of-Squares Proofs" @default.
- W4299626717 doi "https://doi.org/10.48550/arxiv.1504.01656" @default.
- W4299626717 hasPublicationYear "2015" @default.
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